DOCSLIB.ORG

Linear and Quadratic Discriminant Analysis: Tutorial

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear and Quadratic Discriminant Analysis: Tutorial Linear and Quadratic Discriminant Analysis: Tutorial Benyamin Ghojogh [email protected] Department of Electrical and Computer Engineering, Machine Learning Laboratory, University of Waterloo, Waterloo, ON, Canada Mark Crowley [email protected] Department of Electrical and Computer Engineering, Machine Learning Laboratory, University of Waterloo, Waterloo, ON, Canada Abstract fundamental methods. Finally, some experiments on syn- This tutorial explains Linear Discriminant Anal- thetic datasets are reported and analyzed for illustration. ysis (LDA) and Quadratic Discriminant Analysis (QDA) as two fundamental classification meth- 2. Optimization for the Boundary of Classes ods in statistical and probabilistic learning. We First suppose the data is one dimensional, x 2 R. As- start with the optimization of decision boundary sume we have two classes with the Cumulative Distribu- on which the posteriors are equal. Then, LDA tion Functions (CDF) F1(x) and F2(x), respectively. Let and QDA are derived for binary and multiple the Probability Density Functions (PDF) of these CDFs be: classes. The estimation of parameters in LDA and QDA are also covered. Then, we explain @F (x) f (x) = 1 ; (1) how LDA and QDA are related to metric learn- 1 @x ing, kernel principal component analysis, Maha- @F (x) f (x) = 2 ; (2) lanobis distance, logistic regression, Bayes op- 2 @x timal classifier, Gaussian naive Bayes, and like- lihood ratio test. We also prove that LDA and respectively. Fisher discriminant analysis are equivalent. We We assume that the two classes have normal (Gaussian) finally clarify some of the theoretical concepts distribution which is the most common and default distri- with simulations we provide. bution in the real-world applications. The mean of one of the two classes is greater than the other one; we assume µ1 < µ2. An instance x 2 R belongs to one of these two 1. Introduction classes: n Assume we have a dataset of instances f(xi; yi)gi=1 with 2 d N (µ1; σ1); if x 2 C1; sample size n and dimensionality xi 2 R and yi 2 R. The x ∼ 2 (3) N (µ2; σ2); if x 2 C2; yi’s are the class labels. We would like to classify the space of data using these instances. Linear Discriminant Analysis where C and C denote the first and second class, respec- (LDA) and Quadratic discriminant Analysis (QDA) (Fried- 1 2 arXiv:1906.02590v1 [stat.ML] 1 Jun 2019 tively. man et al., 2009) are two well-known supervised classifica- x tion methods in statistical and probabilistic learning. This For an instance , we may have an error in estimation of x∗ paper is a tutorial for these two classifiers where the the- the class it belongs to. At a point, which we denote by , ory for binary and multi-class classification are detailed. the probability of the two classes are equal; therefore, the x∗ Then, relations of LDA and QDA to metric learning, ker- point is on the boundary of the two classes. As we have µ < µ µ < x∗ < µ nel Principal Component Analysis (PCA), Fisher Discrim- 1 2, we can say 1 2 as shown in Fig.1. x < x∗ x > x∗ x inant Analysis (FDA), logistic regression, Bayes optimal Therefore, if or the instance belongs to classifier, Gaussian naive Bayes, and Likelihood Ratio Test the first and second class, respectively. Hence, estimating x < x∗ x > x∗ (LRT) are explained for better understanding of these two or to respectively belong to the second and first class is an error in estimation of the class. This probability of the error can be stated as: ∗ ∗ P(error) = P(x > x ; x 2 C1) + P(x < x ; x 2 C2): (4) Linear and Quadratic Discriminant Analysis: Tutorial 2 where jCj is the number of classes which is two here. The f1(x) and π1 are the likelihood (class conditional) and prior probabilities, respectively, and the denominator is the marginal probability. Therefore, Eq. (13) becomes: f1(x) π1 PjCj (X = x j x 2 Ci) πi Figure 1. Two Gaussian density functions where they are equal at i=1 P ∗ f (x) π the point x . =set 2 2 PjCj i=1 P(X = x j x 2 Ci) πi As we have P(A; B) = P(AjB) P(B), we can say: =) f1(x) π1 = f2(x) π2: (15) ∗ P(error) = P(x > x j x 2 C1) P(x 2 C1) ∗ (5) + P(x < x j x 2 C2) P(x 2 C2); which we want to minimize: Now let us think of data as multivariate data with dimen- sionality d. The PDF for multivariate Gaussian distribu- minimize (error); (6) x∗ P tion, x ∼ N (µ; Σ) is: by finding the best boundary of classes, i.e., x∗. According to the definition of CDF, we have: > −1 (x < c; x 2 C ) = F (c); 1 (x − µ) Σ (x − µ) P 1 1 f(x) = p exp − ; ∗ ∗ (2π)djΣj 2 =) P(x > x ; x 2 C1) = 1 − F1(x ); (7) ∗ ∗ (16) P(x < x ; x 2 C2) = F2(x ): (8) According to the definition of PDF, we have: (x 2 C ) = f (x) = π ; (9) d d d×d P 1 1 1 where x 2 R , µ 2 R is the mean, Σ 2 R is the P(x 2 C2) = f2(x) = π2; (10) covariance matrix, and j:j is the determinant of matrix. The π ≈ 3:14 in this equation should not be confused with the where we denote the priors f (x) and f (x) by π and π , 1 2 1 2 π (prior) in Eq. (12) or (15). Therefore, the Eq. (12) or respectively. k (15) becomes: Hence, Eqs. (5) and (6) become: ∗ ∗ minimize 1 − F1(x ) π1 + F2(x ) π2: (11) x∗ > −1 We take derivative for the sake of minimization: 1 (x − µ1) Σ1 (x − µ1) exp − π1 p d 2 @ P(error) ∗ ∗ set (2π) jΣ1j = −f1(x ) π1 + f2(x ) π2 = 0; @x∗ > −1 1 (x − µ2) Σ2 (x − µ2) ∗ ∗ = exp − π2; =) f1(x ) π1 = f2(x ) π2: (12) p d (2π) jΣ2j 2 Another way to obtain this expression is equating the pos- (17) terior probabilities to have the equation of the boundary of where the distributions of the first and second class are classes: N (µ1; Σ1) and N (µ2; Σ2), respectively. set P(x 2 C1 j X = x) = P(x 2 C2 j X = x): (13) 3. Linear Discriminant Analysis for Binary According to Bayes rule, the posterior is: Classification P(X = x j x 2 C1) P(x 2 C1) In Linear Discriminant Analysis (LDA), we assume that the P(x 2 C1 j X = x) = two classes have equal covariance matrices: P(X = x) f (x) π = 1 1 ; PjCj k=1 P(X = x j x 2 Ck) πk (14) Σ1 = Σ2 = Σ: (18) Linear and Quadratic Discriminant Analysis: Tutorial 3 Therefore, the Eq. (17) becomes: the class of an instance x is estimated as: > −1 1; if δ(x) < 0; 1 (x − µ1) Σ (x − µ1) Cb(x) = (22) exp − π1 2; if δ(x) > 0: p(2π)djΣj 2 > −1 If the priors of two classes are equal, i.e., π1 = π2, the Eq. 1 (x − µ2) Σ (x − µ2) = exp − π2; (20) becomes: p(2π)djΣj 2 −1 > > −1 2 Σ (µ − µ ) x (x − µ1) Σ (x − µ1) 2 1 (23) =) exp − π1 > −1 2 + µ1 − µ2) Σ (µ1 − µ2) = 0; > −1 (x − µ2) Σ (x − µ2) = exp − π2; whose left-hand-side expression can be considered as δ(x) 2 in Eq. (22). (a) 1 > −1 =) − (x − µ1) Σ (x − µ1) + ln(π1) 2 4. Quadratic Discriminant Analysis for 1 > −1 = − (x − µ ) Σ (x − µ ) + ln(π2); Binary Classification 2 2 2 In Quadratic Discriminant Analysis (QDA), we relax the where (a) takes natural logarithm from the sides of equa- assumption of equality of the covariance matrices: tion. We can simplify this term as: Σ1 6= Σ2; (24) which means the covariances are not necessarily equal (if (x − µ )>Σ−1(x − µ ) = (x> − µ>)Σ−1(x − µ ) 1 1 1 1 they are actually equal, the decision boundary will be linear > −1 > −1 > −1 > −1 = x Σ x − x Σ µ1 − µ1 Σ x + µ1 Σ µ1 and QDA reduces to LDA). (a) > −1 > −1 > −1 Therefore, the Eq. (17) becomes: = x Σ x + µ1 Σ µ1 − 2 µ1 Σ x; (19) > −1 1 (x − µ1) Σ1 (x − µ1) where (a) is because x>Σ−1µ = µ>Σ−1x as it is a exp − π1 1 1 p(2π)djΣ j 2 scalar and Σ−1 is symmetric so Σ−> = Σ−1. Thus, we 1 > −1 have: 1 (x − µ2) Σ2 (x − µ2) = exp − π2; p d (2π) jΣ2j 2 1 > −1 1 > −1 > −1 − x Σ x − µ1 Σ µ1 + µ1 Σ x + ln(π1) (a) d 1 2 2 =) − ln(2π) − ln(jΣ1j) 1 > −1 1 > −1 > −1 2 2 = − x Σ x − µ2 Σ µ2 + µ2 Σ x + ln(π2): 1 2 2 − (x − µ )>Σ−1(x − µ ) + ln(π ) 2 1 1 1 1 Therefore, if we multiply the sides of equation by 2, we d 1 = − ln(2π) − ln(jΣ j) have: 2 2 2 > 1 2 Σ−1(µ − µ ) x − (x − µ )>Σ−1(x − µ ) + ln(π ); 2 1 2 2 2 2 2 > −1 π2 + µ1 − µ2) Σ (µ1 − µ2) + 2 ln( ) = 0; where (a) takes natural logarithm from the sides of equa- π1 (20) tion. According to Eq. (19), we have: > which is the equation of a line in the form of a x + b = 0.
Load more

Recommended publications
  • Solving Cubic Polynomials
    Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac.
    [Show full text]
  • Elements of Chapter 9: Nonlinear Systems Examples
    Elements of Chapter 9: Nonlinear Systems To solve x0 = Ax, we use the ansatz that x(t) = eλtv. We found that λ is an eigenvalue of A, and v an associated eigenvector. We can also summarize the geometric behavior of the solutions by looking at a plot- However, there is an easier way to classify the stability of the origin (as an equilibrium), To find the eigenvalues, we compute the characteristic equation: p Tr(A) ± ∆ λ2 − Tr(A)λ + det(A) = 0 λ = 2 which depends on the discriminant ∆: • ∆ > 0: Real λ1; λ2. • ∆ < 0: Complex λ = a + ib • ∆ = 0: One eigenvalue. The type of solution depends on ∆, and in particular, where ∆ = 0: ∆ = 0 ) 0 = (Tr(A))2 − 4det(A) This is a parabola in the (Tr(A); det(A)) coordinate system, inside the parabola is where ∆ < 0 (complex roots), and outside the parabola is where ∆ > 0. We can then locate the position of our particular trace and determinant using the Poincar´eDiagram and it will tell us what the stability will be. Examples Given the system where x0 = Ax for each matrix A below, classify the origin using the Poincar´eDiagram: 1 −4 1. 4 −7 SOLUTION: Compute the trace, determinant and discriminant: Tr(A) = −6 Det(A) = −7 + 16 = 9 ∆ = 36 − 4 · 9 = 0 Therefore, we have a \degenerate sink" at the origin. 1 2 2. −5 −1 SOLUTION: Compute the trace, determinant and discriminant: Tr(A) = 0 Det(A) = −1 + 10 = 9 ∆ = 02 − 4 · 9 = −36 The origin is a center. 1 3. Given the system x0 = Ax where the matrix A depends on α, describe how the equilibrium solution changes depending on α (use the Poincar´e Diagram): 2 −5 (a) α −2 SOLUTION: The trace is 0, so that puts us on the \det(A)" axis.
    [Show full text]
  • Polynomials and Quadratics
    Higher hsn .uk.net Mathematics UNIT 2 OUTCOME 1 Polynomials and Quadratics Contents Polynomials and Quadratics 64 1 Quadratics 64 2 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining the Equation of a Parabola 72 6 Solving Quadratic Inequalities 74 7 Intersections of Lines and Parabolas 76 8 Polynomials 77 9 Synthetic Division 78 10 Finding Unknown Coefficients 82 11 Finding Intersections of Curves 84 12 Determining the Equation of a Curve 86 13 Approximating Roots 88 HSN22100 This document was produced specially for the HSN.uk.net website, and we require that any copies or derivative works attribute the work to Higher Still Notes. For more details about the copyright on these notes, please see http://creativecommons.org/licenses/by-nc-sa/2.5/scotland/ Higher Mathematics Unit 2 – Polynomials and Quadratics OUTCOME 1 Polynomials and Quadratics 1 Quadratics A quadratic has the form ax2 + bx + c where a, b, and c are any real numbers, provided a ≠ 0 . You should already be familiar with the following. The graph of a quadratic is called a parabola . There are two possible shapes: concave up (if a > 0 ) concave down (if a < 0 ) This has a minimum This has a maximum turning point turning point To find the roots (i.e. solutions) of the quadratic equation ax2 + bx + c = 0, we can use: factorisation; completing the square (see Section 3); −b ± b2 − 4 ac the quadratic formula: x = (this is not given in the exam). 2a EXAMPLES 1. Find the roots of x2 −2 x − 3 = 0 .
    [Show full text]
  • Quadratic Polynomials
    Quadratic Polynomials If a>0thenthegraphofax2 is obtained by starting with the graph of x2, and then stretching or shrinking vertically by a. If a<0thenthegraphofax2 is obtained by starting with the graph of x2, then flipping it over the x-axis, and then stretching or shrinking vertically by the positive number a. When a>0wesaythatthegraphof− ax2 “opens up”. When a<0wesay that the graph of ax2 “opens down”. I Cit i-a x-ax~S ~12 *************‘s-aXiS —10.? 148 2 If a, c, d and a = 0, then the graph of a(x + c) 2 + d is obtained by If a, c, d R and a = 0, then the graph of a(x + c)2 + d is obtained by 2 R 6 2 shiftingIf a, c, the d ⇥ graphR and ofaax=⇤ 2 0,horizontally then the graph by c, and of a vertically(x + c) + byd dis. obtained (Remember by shiftingshifting the the⇥ graph graph of of axax⇤ 2 horizontallyhorizontally by by cc,, and and vertically vertically by by dd.. (Remember (Remember thatthatd>d>0meansmovingup,0meansmovingup,d<d<0meansmovingdown,0meansmovingdown,c>c>0meansmoving0meansmoving thatleft,andd>c<0meansmovingup,0meansmovingd<right0meansmovingdown,.) c>0meansmoving leftleft,and,andc<c<0meansmoving0meansmovingrightright.).) 2 If a =0,thegraphofafunctionf(x)=a(x + c) 2+ d is called a parabola. If a =0,thegraphofafunctionf(x)=a(x + c)2 + d is called a parabola. 6 2 TheIf a point=0,thegraphofafunction⇤ ( c, d) 2 is called thefvertex(x)=aof(x the+ c parabola.) + d is called a parabola. The point⇤ ( c, d) R2 is called the vertex of the parabola.
    [Show full text]
  • The Determinant and the Discriminant
    CHAPTER 2 The determinant and the discriminant In this chapter we discuss two indefinite quadratic forms: the determi- nant quadratic form det(a, b, c, d)=ad bc, − and the discriminant disc(a, b, c)=b2 4ac. − We will be interested in the integral representations of a given integer n by either of these, that is the set of solutions of the equations 4 ad bc = n, (a, b, c, d) Z − 2 and 2 3 b ac = n, (a, b, c) Z . − 2 For q either of these forms, we denote by Rq(n) the set of all such represen- tations. Consider the three basic questions of the previous chapter: (1) When is Rq(n) non-empty ? (2) If non-empty, how large Rq(n)is? (3) How is the set Rq(n) distributed as n varies ? In a suitable sense, a good portion of the answers to these question will be similar to the four and three square quadratic forms; but there will be major di↵erences coming from the fact that – det and disc are indefinite quadratic forms (have signature (2, 2) and (2, 1) over the reals), – det and disc admit isotropic vectors: there exist x Q4 (resp. Q3) such that det(x)=0(resp.disc(x) = 0). 2 1. Existence and number of representations by the determinant As the name suggest, determining Rdet(n) is equivalent to determining the integral 2 2 matrices of determinant n: ⇥ (n) ab Rdet(n) M (Z)= g = M2(Z), det(g)=n . ' 2 { cd 2 } ✓ ◆ n 0 Observe that the diagonal matrix a = has determinant n, and any 01 ✓ ◆ other matrix in the orbit SL2(Z).a is integral and has the same determinant.
    [Show full text]
  • Nature of the Discriminant
    Name: ___________________________ Date: ___________ Class Period: _____ Nature of the Discriminant Quadratic − b b 2 − 4ac x = b2 − 4ac Discriminant Formula 2a The discriminant predicts the “nature of the roots of a quadratic equation given that a, b, and c are rational numbers. It tells you the number of real roots/x-intercepts associated with a quadratic function. Value of the Example showing nature of roots of Graph indicating x-intercepts Discriminant b2 – 4ac ax2 + bx + c = 0 for y = ax2 + bx + c POSITIVE Not a perfect x2 – 2x – 7 = 0 2 b – 4ac > 0 square − (−2) (−2)2 − 4(1)(−7) x = 2(1) 2 32 2 4 2 x = = = 1 2 2 2 2 Discriminant: 32 There are two real roots. These roots are irrational. There are two x-intercepts. Perfect square x2 + 6x + 5 = 0 − 6 62 − 4(1)(5) x = 2(1) − 6 16 − 6 4 x = = = −1,−5 2 2 Discriminant: 16 There are two real roots. These roots are rational. There are two x-intercepts. ZERO b2 – 4ac = 0 x2 – 2x + 1 = 0 − (−2) (−2)2 − 4(1)(1) x = 2(1) 2 0 2 x = = = 1 2 2 Discriminant: 0 There is one real root (with a multiplicity of 2). This root is rational. There is one x-intercept. NEGATIVE b2 – 4ac < 0 x2 – 3x + 10 = 0 − (−3) (−3)2 − 4(1)(10) x = 2(1) 3 − 31 3 31 x = = i 2 2 2 Discriminant: -31 There are two complex/imaginary roots. There are no x-intercepts. Quadratic Formula and Discriminant Practice 1.
    [Show full text]
  • Resultant and Discriminant of Polynomials
    RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results are well-known 19th century mathematics, but I have not inves- tigated the history, and no references are given. 1. Resultant n m Definition 1.1. Let f(x) = anx + ··· + a0 and g(x) = bmx + ··· + b0 be two polynomials of degrees (at most) n and m, respectively, with coefficients in an arbitrary field F . Their resultant R(f; g) = Rn;m(f; g) is the element of F given by the determinant of the (m + n) × (m + n) Sylvester matrix Syl(f; g) = Syln;m(f; g) given by 0an an−1 an−2 ::: 0 0 0 1 B 0 an an−1 ::: 0 0 0 C B . C B . C B . C B C B 0 0 0 : : : a1 a0 0 C B C B 0 0 0 : : : a2 a1 a0C B C (1.1) Bbm bm−1 bm−2 ::: 0 0 0 C B C B 0 bm bm−1 ::: 0 0 0 C B . C B . C B C @ 0 0 0 : : : b1 b0 0 A 0 0 0 : : : b2 b1 b0 where the m first rows contain the coefficients an; an−1; : : : ; a0 of f shifted 0; 1; : : : ; m − 1 steps and padded with zeros, and the n last rows contain the coefficients bm; bm−1; : : : ; b0 of g shifted 0; 1; : : : ; n−1 steps and padded with zeros. In other words, the entry at (i; j) equals an+i−j if 1 ≤ i ≤ m and bi−j if m + 1 ≤ i ≤ m + n, with ai = 0 if i > n or i < 0 and bi = 0 if i > m or i < 0.
    [Show full text]
  • Lyashko-Looijenga Morphisms and Submaximal Factorisations of A
    LYASHKO-LOOIJENGA MORPHISMS AND SUBMAXIMAL FACTORIZATIONS OF A COXETER ELEMENT VIVIEN RIPOLL Abstract. When W is a finite reflection group, the noncrossing partition lattice NC(W ) of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NC(W ) as a generalized Fuß-Catalan number, depending on the invariant degrees of W . We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NC(W ) as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W . We study algebraically the map LL, describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of W . Introduction Complex reflection groups are a natural generalization of finite real reflection groups (that is, finite Coxeter groups realized in their geometric representation). In this article, we consider a well-generated complex reflection group W ; the precise definitions will be given in Sec. 1.1. The noncrossing partition lattice of type W , denoted NC(W ), is a particular subset of W , endowed with a partial order 4 called the absolute order (see definition below). When W is a Coxeter group of type A, NC(W ) is isomorphic to the poset of noncrossing partitions of a set, studied by Kreweras [Kre72].
    [Show full text]
  • Galois Groups of Cubics and Quartics (Not in Characteristic 2)
    GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) KEITH CONRAD We will describe a procedure for figuring out the Galois groups of separable irreducible polynomials in degrees 3 and 4 over fields not of characteristic 2. This does not include explicit formulas for the roots, i.e., we are not going to derive the classical cubic and quartic formulas. 1. Review Let K be a field and f(X) be a separable polynomial in K[X]. The Galois group of f(X) over K permutes the roots of f(X) in a splitting field, and labeling the roots as r1; : : : ; rn provides an embedding of the Galois group into Sn. We recall without proof two theorems about this embedding. Theorem 1.1. Let f(X) 2 K[X] be a separable polynomial of degree n. (a) If f(X) is irreducible in K[X] then its Galois group over K has order divisible by n. (b) The polynomial f(X) is irreducible in K[X] if and only if its Galois group over K is a transitive subgroup of Sn. Definition 1.2. If f(X) 2 K[X] factors in a splitting field as f(X) = c(X − r1) ··· (X − rn); the discriminant of f(X) is defined to be Y 2 disc f = (rj − ri) : i<j In degree 3 and 4, explicit formulas for discriminants of some monic polynomials are (1.1) disc(X3 + aX + b) = −4a3 − 27b2; disc(X4 + aX + b) = −27a4 + 256b3; disc(X4 + aX2 + b) = 16b(a2 − 4b)2: Theorem 1.3.
    [Show full text]
  • Discriminants, Resultants, and Their Tropicalization
    Discriminants, resultants, and their tropicalization Course by Bernd Sturmfels - Notes by Silvia Adduci 2006 Contents 1 Introduction 3 2 Newton polytopes and tropical varieties 3 2.1 Polytopes . 3 2.2 Newton polytope . 6 2.3 Term orders and initial monomials . 8 2.4 Tropical hypersurfaces and tropical varieties . 9 2.5 Computing tropical varieties . 12 2.5.1 Implementation in GFan ..................... 12 2.6 Valuations and Connectivity . 12 2.7 Tropicalization of linear spaces . 13 3 Discriminants & Resultants 14 3.1 Introduction . 14 3.2 The A-Discriminant . 16 3.3 Computing the A-discriminant . 17 3.4 Determinantal Varieties . 18 3.5 Elliptic Curves . 19 3.6 2 × 2 × 2-Hyperdeterminant . 20 3.7 2 × 2 × 2 × 2-Hyperdeterminant . 21 3.8 Ge’lfand, Kapranov, Zelevinsky . 21 1 4 Tropical Discriminants 22 4.1 Elliptic curves revisited . 23 4.2 Tropical Horn uniformization . 24 4.3 Recovering the Newton polytope . 25 4.4 Horn uniformization . 29 5 Tropical Implicitization 30 5.1 The problem of implicitization . 30 5.2 Tropical implicitization . 31 5.3 A simple test case: tropical implicitization of curves . 32 5.4 How about for d ≥ 2 unknowns? . 33 5.5 Tropical implicitization of plane curves . 34 6 References 35 2 1 Introduction The aim of this course is to introduce discriminants and resultants, in the sense of Gel’fand, Kapranov and Zelevinsky [6], with emphasis on the tropical approach which was developed by Dickenstein, Feichtner, and the lecturer [3]. This tropical approach in mathematics has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields.
    [Show full text]
  • Eisenstein and the Discriminant
    Extra handout: The discriminant, and Eisenstein’s criterion for shifted polynomials Steven Charlton, Algebra Tutorials 2015–2016 http://maths.dur.ac.uk/users/steven.charlton/algebra2_1516/ Recall the Eisenstein criterion from Lecture 8: n Theorem (Eisenstein criterion). Let f(x) = anx + ··· + a1x + a0 ∈ Z[x] be a polynomial. Suppose there is a prime p ∈ Z such that • p | ai, for 0 ≤ i ≤ n − 1, • p - an, and 2 • p - a0. Then f(x) is irreducible in Q[x]. p−1 p−2 You showed the cyclotomic polynomial Φp(X) = x + x + ··· + x + 1 is irreducible by applying Eisenstein to the shift Φp(x + 1). Tutorial 2, Question 1 asks you to find a shift of x2 + x + 2, for which Eisenstein works. How can we tell what shifts to use, and what primes might work? We can use the discriminant of a polynomial. 1 Discriminant n The discriminant of a polynomial f(x) = anx + ··· + a1x + a0 ∈ C[x] is a number ∆(f), which gives some information about the nature of the roots of f. It can be defined as n 2n−2 Y 2 ∆(f) := an (ri − rj) , 1≤i<j≤n where ri are the roots of f(x). Example. 1) Consider the quadratic polynomial f(x) = ax2 + bx + c. The roots of this polynomial are √ 1 2 r1, r2 = a −b ± b − 4ac , √ 1 2 so r1 − r2 = a b − 4ac. We get that the discriminant is √ 2 2·2−2 1 2 2 ∆(f) = a a b − 4ac = b − 4ac , as you probably well know. 1 2) The discriminant of a cubic polynomial f(x) = ax3 + bx2 + cx + d is given by ∆(f) = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd .
    [Show full text]
  • On Facets of the Newton Polytope for the Discriminant of the Polynomial System
    On facets of the Newton polytope for the discriminant of the polynomial system Irina Antipova,∗ Ekaterina Kleshkova† Abstract We study normal directions to facets of the Newton polytope of the discriminant of the Laurent polynomial system via the tropical approach. We use the combinato- rial construction proposed by Dickenstein, Feichtner and Sturmfels for the tropical- ization of algebraic varieties admitting a parametrization by a linear map followed by a monomial map. Keywords: discriminant, Newton polytope, tropical variety, Bergman fan, matroid. 2020 Mathematical Subject Classification: 14M25, 14T15, 32S25. 1 Introduction Consider a system of n polynomial equations of the form X (i) λ Pi := aλ y = 0; i = 1; : : : ; n (1) λ2A(i) n (i) with unknowns y = (y1; : : : ; yn) 2 (C n 0) , variable complex coefficients a = (aλ ), where (i) n the sets A ⊂ Z are fixed, finite and contain the zero element 0, λ = (λ1; : : : ; λn), λ λ1 λn y = y1 · ::: · yn . Solution y(a) = (y1(a); : : : ; yn(a)) of (1) has a polyhomogeneity property, and thus the system usually can be written in a homogenized form by means of monomial transformations x = x(a) of coefficients (see [1]). To do this, it is necessary to distinguish a collection of n exponents !(i) 2 A(i) such that the matrix ! = !(1);:::;!(n) is non-degenerated. As a result, we obtain a reduced system of the form !(i) X (i) λ Qi := y + xλ y − 1 = 0; i = 1; : : : ; n; (2) arXiv:2107.04978v1 [math.AG] 11 Jul 2021 λ2Λ(i) (i) (i) (i) (i) where Λ := A n f! ; 0g and x = (xλ ) are variable complex coefficients.
    [Show full text]